It is always worth removing all the automatic and control everything manually.
Reset the AE to factory and redo the setup.. follow these rules. It is not all required but it can help track down the issue.
Factory reset universal
Power off the AE.. ie pull the power cord or power off at the wall.. wait 10sec.. hold in the reset button.. be gentle.. power on again still holding in reset.. and keep holding it in for another 10sec. You may need some help as it is hard to both hold in reset and apply power. It will show success by rapidly blinking the front led. Release the reset.. and wait a couple of min for the AE to reset and come back with factory settings. If the front LED doesn’t blink rapidly you missed it and simply try again. The reset is fairly fragile in these.. press it so you feel it just click and no more.. I have seen people bend the lever or even break it. I use a toothpick as tool.
Setup the AE again.
Setup from the computer with Mavericks--Yosemite--ElCapo. (MYELCAP)
1. Use very short names.. NOT APPLE RECOMMENDED names. No spaces and pure alphanumerics.
eg AEgen5 for basestation and AEwifi wireless name.
If the issue is wireless use AE4ghz and AE5ghz with fixed channels as this also seems to help stop the nonsense. But this can be tried in the second round. ie plan on a first and second round of changes to fix this.. hopefully.. I will point out other steps that can be round2.
2. Use all passwords that also comply with 1. but can be a bit longer. ie 8-20 characters mixed case and numbers.. no non-alphanumerics.
3. If the AE is main router you can skip this point. This is only an issue when the AE is bridged.
Ensure the AE always takes the same IP address.. you will need to do this on the main router using dhcp reservation.. or a bit more complex setup using static IP in the AE. But this is important.. having IP drift all over the place when MYELCAP cannot remember its own name for 5 min after a reboot makes for poor networking.
If you don't understand this one.. ignore it for now.. but it will come back to bite you in the future so ask.
4. Check your share name on the computer is not changing.. make sure it also complies with the above.. short no spaces and pure alphanumeric.. but this change will mess up your TM backup.. so be prepared to do a new full backup. Sorry.. keep this one for second round if you want to avoid a new backup.
5. If you have hard disk plugged into the AE.. Mount the AE disk in the computer manually.
In Finder, Go, Connect to server from the top menu,
Type in SMB://192.168.0.254 (or whatever the AE ip is which you have now made static. As a router by default it is 10.0.1.1 and I encourage people to stick with that unless you know what you are doing).
You can use name.. SMB://AEgen6.local where you replace AEgen6 with your AE name.. local is the default domain of the AE and doesn't change.
However names are not so easy as IP address.. nor as reliable. At least not in Yosemite they aren't. The domain can also be an issue if you are not plugged or wireless directly to the AE.
6. Make sure IPv6 is set to link-local only in the computer. For example wireless open the network preferences, wireless and advanced / TCP/IP.. and fix the IPv6. to link-local only. Do the same for ethernet if you use it.
There is a lot more jiggery pokery you can try but the above is a good start.. if you find it still unreliable.. don't be surprised.
You might need to do some more work on the computer itself. eg Reset the NVRAM/PRAM/SMC.. has helped some people. Clean install of the OS is also helpful if you upgrade installed.
https://discussions.apple.com/thread/7306452?answerId=29202159022#29202159022&ac_cid=ha123456#29202159
It is hard to believe that it just failed out of the blue. It is less than a year old.
The AC model extreme has proven somewhat unreliable.. especially if you have the wrong cable modem router setup.. it gets very fussy.